Method and apparatus for discrete multitone transmission

ABSTRACT

A method for discrete multitone (DMT) transmission is disclosed. In the method, a DMT signal is received from a transmission channel. The DMT signal is passed through a time-domain equalizer (TEQ) to obtain an equalized DMT signal. The DMT signal is passed through a target impulse response (TIR) filter to obtain a TIR signal. A mean square error (MSE) of an error signal is obtained from the equalized DMT signal and the TIR signal. A TEQ coefficient vector of the TEQ is iteratively updated based on the MSE of the error signal, a frequency kernel matrix corresponding to the TEQ and a frequency kernel matrix corresponding to the TIR filter.

BACKGROUND

Field of the Invention

The present invention relates to a transmission system, and moreparticularly to a method and apparatus of a discrete multitone (DMT)transmission system.

Description of Related Art

DMT modulation is a form of multicarrier modulation that divides theavailable bandwidth into several independent subchannels. The DMTmodulation is able to adapt bit and power allocation of each subchannel,such that the throughput of each subchannel is maximized. Among thesubchannels, if a subchannel is unable to be used for transmission dueto serious interference from external environment, the subchannel can beturned off, while the other subchannels are not affected, such that theavailable bandwidth is optimized. With at least these advantages, DMTtransmission is extensively used in broadband wireline communicationsystems, such as asymmetric digital subscriber line (ADSL) and very-highspeed digital subscriber line (VDSL) systems. The DMT transmission isalso proposed as a potential solution in the next generationserializer-deserializer (SERDES) system with a signal throughput up to56 Gbps or 112 Gbps.

SUMMARY

The objective of the present invention is to provide a method andapparatus for DMT transmission. By utilizing the method and apparatus ofthe present invention for a DMT transmission system, both of thetime-domain response and the frequency-domain response of thetime-domain equalizer (TEQ) in the null band can be effectivelysuppressed, thereby improving the DMT transmission quality.

One aspect of the present invention is to provide a method for discretemultitone (DMT) transmission. In this method, a DMT signal is receivedfrom a transmission channel. The received DMT signal is passed throughthe TEQ to obtain an equalized DMT signal. The DMT signal is passedthrough a target impulse response (TIR) filter to obtain a TIR signal. Amean square error (MSE) of an error signal is obtained from theequalized DMT signal and the TIR signal. A TEQ coefficient vector of theTEQ is iteratively updated based on the MSE of the error signal, afrequency kernel matrix corresponding to the TEQ and a frequency kernelmatrix corresponding to the TIR filter.

Another aspect of the present invention is to provide an apparatus forDMT transmission. The apparatus includes a TEQ, a TIR filter, an adderand a processor. The TEQ is configured to pass a received DMT signaltherethrough to obtain an equalized DMT signal. The TIR filter isconfigured to pass the DMT signal therethrough to obtain a TIR signal.The adder is configured to generate an error signal from the equalizedDMT signal and the TIR signal. The processor is configured to performoperations including obtaining a MSE of an error signal from theequalized DMT signal and the TIR signal and iteratively updating a TEQcoefficient vector of the TEQ based on the MSE of the error signal, afrequency kernel matrix corresponding to the TEQ and a frequency kernelmatrix corresponding to the TIR filter.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention can be more fully understood by reading thefollowing detailed description of the embodiment, with reference made tothe accompanying drawings as follows:

FIG. 1 illustrates a block diagram of a DMT transmission system inaccordance with some embodiments of the present invention.

FIG. 2 illustrates a minimum mean square error (MMSE) system model usedfor the DMT transmission system of FIG. 1 for determining TEQ filtercoefficients in accordance with some embodiments of the presentinvention.

FIG. 3 illustrates a frequency spectrum for a DMT-based transmissionsystem.

FIG. 4 shows a frequency-domain response of the TEQ in FIG. 1 with theTEQ coefficients calculated by iteratively updating an optimum TEQcoefficient vector obtained from a cost function.

FIG. 5 shows a frequency-domain response of the TEQ in FIG. 1 with theTEQ coefficients calculated by iteratively updating an optimum TEQcoefficient vector obtained from a modified cost function in accordancewith some embodiments of the present invention.

DETAILED DESCRIPTION

Reference will now be made in detail to the embodiments of the presentinvention, examples of which are illustrated in the accompanyingdrawings. Wherever possible, the same reference numbers are used in thedrawings and the description to refer to the same or like parts.

FIG. 1 illustrates a block diagram of a discrete multitone (DMT)transmission system 100 in accordance with some embodiments of thepresent invention. The DMT transmission system 100 implements DMT datatransmission technology, and has been standardized for various types ofdigital subscriber lines (DSLs) transmission, such as asymmetric digitalsubscriber lines (ADSLs) transmission, very-high-bit-rate digitalsubscriber lines (VDSLs) transmission.

As shown in FIG. 1, the DMT transmission system 100 includes atransmitter 110, a receiver 120 and a transmission channel 130 betweenthe transmitter 110 and the receiver 120. The transmitter 110 convertsinput binary data into DMT signal, and the DMT signal is transmitted tothe receiver 120 through the transmission channel 130.

The transmitter 110 includes a serial-to-parallel (S/P) converter 111, aconstellation mapper 112, an inverse digital Fourier transformer (IDFT)113, a parallel-to-serial (P/S) converter 114 and a cyclic prefix (CP)generator 115. The S/P converter 111 demultiplexes the input binary datafrom serial form into parallel form. The constellation mapper 112 mapsthe input binary data into a complex number for each subchannel. TheIDFT 113 transforms the mapped complex numbers of all suubchannels fromfrequency-domain into time-domain. The IDFT 113 may use an inverse FastFourier Transform (IFFT) algorithm to implement the frequency-domain totime-domain transformation. The P/S converter 114 converts the paralleltime-domain input data into serial time-domain output samples, and theserial time-domain output samples comprise DMT symbols. The CP generator115 inserts cyclic prefixes into the serial time-domain output samplesto form the DMT signal. The transmitter 110 further includes adigital-to-analog (D/A) converter (not shown) for converting the DMTsignal into analog form labeled as x(i).

The DMT signal x(i) is transmitted to the receiver 120 through thetransmission channel 130 with a channel impulse response h(i) and theadder 140 where an additive noise n(i) is added thereto. Therelationship between the received DMT signal r(i), the DMT signal x(i),the channel impulse response h(i) and the additive noise n(i) isr(i)=x(i)*h(i)+n(i), where * denotes a convolution operation.

The receiver 120 includes a time-domain equalizer (TEQ) 121, a CPremover 122, a S/P converter 123, a digital Fourier transformer (DFT)124, a frequency equalizer (FEQ) 125, a constellation demapper 126 and aP/S converter 127. The TEQ 121 equalizes the received DMT signal r(i) toobtain an equalized DMT signal y(i). In addition, the receiver 120further includes an analog-to-digital (A/D) converter (not shown) forconverting the received DMT signal y(i) into analog form beforetransmitting the received DMT signal y(i) to the TEQ 121. The CP remover122 removes the cyclic prefixes from the received DMT signal to generatetime-domain serial data. The S/P converter 123 converts the time-domainserial data into time-domain parallel data. The DFT 124 transforms thetime-domain parallel data into frequency-domain. The DFT 124 may use aFast Fourier Transform (FFT) algorithm to implement the time-domain tofrequency-domain transformation. The FEQ 125 performs single-tapequalization per subcarrier on the frequency-domain parallel data. Theconstellation demapper 126 performs demapping corresponding to theconstellation mapping of the constellation mapper 112 to the equalizedfrequency-domain parallel data outputted by the FEQ 125 to generateparallel output binary data, which is the estimation of the input binarydata in the multibit subchannels. The P/S converter 127 multiplexes theparallel output binary data into serial form.

However, for the DMT transmission system 100, if the memory order of thechannel impulse response h(i) is greater than the length of the cyclicprefixes, undesirable disturbances, such as inter-symbol interference(ISI) and inter-carrier interference (ICI), will be produced, resultingin degradation of signal transmission.

To avoid the ISI and ICI, TEQ filter coefficients of the TEQ 121 aredetermined to shorten the effective length of the transmission channel130. FIG. 2 illustrates a minimum mean square error (MMSE) system modelused for the DMT transmission system 100 of FIG. 1 for determining theTEQ filter coefficients in accordance with some embodiments of thepresent invention. As shown in FIG. 2, the MMSE system model providestwo branches; the first branch includes the transmission channel 130 andthe TEQ 121, and the second branch is a hypothetical parallel branch,which includes a delay channel 210 and a target impulse response (TIR)filter 220. The first branch is the same as the transmission path shownin FIG. 1, and thus the detailed description thereof is not repeated.

In the second branch, the delay channel 210 provides a delay function tothe DMT signal x(i) with a delay Δ via the transmission channel 130 andthe TEQ 121, and the TIR filter 220 filters the delayed DMT signalx(k−Δ) with virtual TIR coefficients to obtain a TIR signal d(i).

The adder 230 generates and outputs an error signal e(i) by subtractingthe TIR signal d(i) from the input signal equalized DMT signal y(i). Thepower of the error signal e(i) is minimized by shortening the channelimpulse response via determining optimum TEQ coefficients for the TEQ121. The optimum TEQ coefficients for minimizing the error signal e(i)can be obtained from minimizing a cost function, which is expressed asEquation (1):E{e ² }=E{w ^(T) ·y−b ^(T) ·x},  (1)where E{e²} is the MSE (mean square error) of the error signal e(i), wis a TEQ coefficient vector of the TEQ 121, y is a sample vector of theequalized DMT signal y(i), x is a sample vector of the transmitted DMTsignal x(i), b is a TIR coefficient vector of the TIR filter 220, and·^(T) is a transpose notation.

To obtain an optimum TEQ coefficient vector w_(opt), let the partialderivative of the MSE E{e²} of the DMT transmission system 100 withrespect to the TEQ coefficient vector w equals 0, and the optimum TEQcoefficient vector w_(opt) can be obtained as:w _(opt) =R _(yy) ⁻¹ R _(yx) b,  (2)where R_(yy) is an autocorrelation matrix of the sample vector y of theequalized DMT signal y(i), and R_(yx) is a cross-correlation matrixbetween the sample vector y and a sample vector x of the transmitted DMTsignal x(i).

FIG. 3 illustrates a frequency spectrum for a DMT-based transmissionsystem, which may be an ADSL system, a VDSL system, a high-speed SERDESsystem, or the like. In FIG. 3, a plain old telephone service (POTS)frequency band, an upstream band and a downstream band are illustrated.The POTS upstream band is usually in the range of 0 to 4 KHz. Theupstream band and the downstream band may be, for example, an ADSLupstream band and an ADSL downstream band, but are not limited thereto.In an ADSL system case, the upstream frequency band is in the range of25.875 KHz to 138 KHz, the download frequency band is in the range of138 KHz to 1.1 MHz. Further, a guard band exists between the POTS andADSL frequency bands.

The overall ADSL frequency band consists of 255 frequency subcarriers(bins) each having a frequency band of 4.3125 KHz. Among the 255frequency subcarriers, 224 frequency subcarriers are in the ADSLdownstream band, and the other 31 frequency subcarriers are in the ADSLupstream band. In some embodiments, some of the frequency subcarriersnear the boundary between the ADSL upstream band and the ADSL downstreamband are used as a guardband.

FIG. 4 shows a frequency-domain response of the TEQ 121 with the TEQcoefficients calculated by iteratively updating the optimum TEQcoefficient vector w_(opt) obtained from Equation (2), where inparticular, the ADSL transmission technology is adopted forillustration. As shown in FIG. 4, for downstream signals with thesubcarriers in the upstream band (null frequency band) are masked byzeros, the optimized TEQ coefficients tend to amplify the magnitude ofthe upstream band (frequency bin number<32). A component of the noisen(i) in the upstream band is amplified, and thus the DMT transmissionquality is affected.

To avoid boosting the null frequency band, the present inventionprovides a modified cost function to obtain optimum TEQ coefficients andTIR coefficients. Specifically, the modified cost function E_(all) isexpressed as Equation (3):E _(all) =E{e ² }+E _(s),  (3)where E{e²} is the MSE of the error signal e(i) obtained from Equation(1), and E_(s) is expressed as Equation (4):E _(s) =wΩ _(w) w ^(T) +bΩ _(b) b ^(T),  (4)where w is the TEQ coefficient vector of the TEQ 121, Ω_(w) is afrequency kernel matrix corresponding to the TEQ 121, b is the TIRcoefficient vector of the TIR filter 220, and Ω_(b) is a frequencykernel matrix corresponding to the TIR filter 220.

In some embodiments, the frequency kernel matrix Ω_(w) is determinedfrom Equation (5):Ω_(w) =w _(l)Ω_(wl) +w _(h)Ω_(wh),  (5)where Ω_(wl) is a low-frequency kernel matrix of the TEQ 121, Ω_(wh) isa high-frequency kernel matrix of the TEQ 121, and w_(l) and w_(h) areweighting factors of the low-frequency kernel matrix Ω_(wl) and thehigh-frequency kernel matrix Ω_(wh), respectively. Similarly, in someembodiments, the frequency kernel matrix Ω_(b) is determined fromEquation (6):Ω_(b) =w _(l)Ω_(bl) +w _(h)Ω_(bh),  (6)where Ω_(bl) is a low-frequency kernel matrix of the TIR filter 220,Ω_(bh) is a high-frequency kernel matrix of the TIR filter 220, andw_(l) and w_(h) are weighting factors of the low-frequency kernel matrixΩ_(bl) and the high-frequency kernel matrix Ω_(bh), respectively. ForEquations (5) and (6), if the low-frequency and high-frequency bands aredetermined in advance, the frequency kernel matrix Ω_(w) and thefrequency kernel matrix Ω_(b) can then be obtained.

In some embodiments, the frequency kernel matrix Ω_(w) may be obtainedby performing an integration operation on a continuous frequency kernelmatrix variable Ω_(w)(ω) with respect to a frequency range of a stopbandof the TEQ 121. For example, for 2× oversampling upstream signals at thereceiver 120 and the TEQ 121 with a normalized frequency range of astopband from π/2 to π, the continuous frequency kernel matrix variableΩ_(w)(ω) is:

$\begin{bmatrix}1 & {\cos(\omega)} & {\cos\left( {2\omega} \right)} & \ldots & {\cos\left\lbrack {\left( {N_{w} - 1} \right)\omega} \right\rbrack} \\{\cos(\omega)} & 1 & {\cos(\omega)} & \ldots & {\cos\left\lbrack {\left( {N_{w} - 2} \right)\omega} \right\rbrack} \\{\cos\left( {2\omega} \right)} & {\cos(\omega)} & 1 & \ldots & {\cos\left\lbrack {\left( {N_{w} - 3} \right)\omega} \right\rbrack} \\\vdots & \vdots & \vdots & \ddots & \vdots \\{\cos\left\lbrack {\left( {N_{w} - 1} \right)\omega} \right\rbrack} & {\cos\left\lbrack {\left( {N_{w} - 2} \right)\omega} \right\rbrack} & {\cos\left\lbrack {\left( {N_{w} - 3} \right)\omega} \right\rbrack} & \ldots & 1\end{bmatrix},$where N_(w) is a TEQ length of the TEQ 121, and ω is the normalizedfrequency. The frequency kernel matrix Ω_(w) is then obtained fromEquation (7):

$\begin{matrix}\begin{matrix}{\Omega_{w} = {\int_{2/\pi}^{\pi}{{\Omega_{w}(\omega)}{\mathbb{d}\omega}}}} \\{= {\begin{bmatrix}{\pi/2} & {- 1} & 0 & \ldots & \frac{\left( {- 1} \right)^{{({N_{w}/2})} - 1}}{N_{w} - 1} \\{- 1} & {\pi/2} & {- 1} & \ldots & 0 \\0 & {- 1} & {\pi/2} & \ldots & \frac{\left( {- 1} \right)^{{({N_{w}/2})} - 3}}{N_{w} - 3} \\\vdots & \vdots & \vdots & \ddots & \vdots \\\frac{\left( {- 1} \right)^{{({N_{w}/2})} - 1}}{N_{w} - 1} & 0 & \frac{\left( {- 1} \right)^{{({N_{w}/2})} - 3}}{N_{w} - 3} & \ldots & {\pi/2}\end{bmatrix}.}}\end{matrix} & (7)\end{matrix}$

Further, in some embodiments, the frequency kernel matrix Ω_(b) may beobtained by performing a summation operation on a discrete frequencykernel matrix variable Ω_(b)(i) with respect to a frequency range of astopband of the TEQ 121. For example, for N_(sb) tones in the stopband,the discrete frequency kernel matrix variable Ω_(b)(i) is:

$\quad{\begin{bmatrix}1 & {\cos\left( \frac{2\pi\; k}{N} \right)} & {\cos\left( \frac{4\pi\; k}{N} \right)} & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 1} \right)\pi\; k}{N} \right\rbrack} \\{\cos\left( \frac{2\pi\; k}{N} \right)} & 1 & {\cos\left( \frac{2\pi\; k}{N} \right)} & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 2} \right)\pi\; k}{N} \right\rbrack} \\{\cos\left( \frac{4\pi\; k}{N} \right)} & {\cos\left( \frac{2\pi\; k}{N} \right)} & 1 & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 3} \right)\pi\; k}{N} \right\rbrack} \\\vdots & \vdots & \vdots & \ddots & \vdots \\{\cos\left\lbrack \frac{2\left( {N_{b} - 1} \right)\pi\; k}{N} \right\rbrack} & {\cos\left\lbrack \frac{2\left( {N_{b} - 2} \right)\pi\; k}{N} \right\rbrack} & {\cos\left\lbrack \frac{2\left( {N_{b} - 3} \right)\pi\; k}{N} \right\rbrack} & \ldots & 1\end{bmatrix},}$where N is the DMT signal length, N_(b)=CP+1, and CP is a cyclic prefixlength. The frequency kernel matrix Ω_(b) is then obtained from Equation(7):

$\begin{matrix}{\Omega_{b} = {\sum\limits_{i = 0}^{N_{s\; b} - 1}{{\Omega_{b}(i)}.}}} & (8)\end{matrix}$

It should be noted that, the frequency kernel matrix Ω_(w) and thefrequency kernel matrix Ω_(b) may be obtained from a discrete orcontinuous matrix variable, but is not limited thereto. That is, thefrequency kernel matrix Ω_(w) may be alternatively obtained from adiscrete frequency kernel matrix variable Ω_(w)(i), and/or the frequencykernel matrix Ω_(b) may be alternatively obtained from a continuousmatrix variable Ω_(b)(ω).

To obtain an optimum TEQ coefficient vector w_(opt), let the partialderivative of the cost function E_(all) of the DMT transmission system100 with respect to the TEQ coefficient vector w equals 0, and theoptimum TEQ coefficient vector w_(opt) can be obtained from Equation(9):w _(opt)=(R _(yy) +kΩ _(w))⁻¹ R _(yx) b,  (9)where R_(yy) is an autocorrelation matrix of a sample vector y of theequalized DMT signal y(i), k is a weighting factor of the frequencykernel matrix Ω_(w), and R_(yx) is a cross-correlation matrix betweenthe sample vector y and a sample vector x of the transmitted DMT signalx(i).

On the other hand, to obtain an optimum TIR coefficient vector b_(opt),let the partial derivative of the cost function E_(all) of the DMTtransmission system 100 with respect to the TIR coefficient vector bequals 0, and the optimum TIR coefficient vector b_(opt) can be obtainedfrom Equation (10):b _(opt)=(R _(xx) +kΩ _(b))⁻¹ R _(xy) w,  (10)where R_(xx) is an autocorrelation matrix of the sample vector x of thetransmitted DMT signal x(i), k is a weighting factor of the frequencykernel matrix Ω_(b), and R_(xy) is a cross-correlation matrix betweenthe sample vector x and the sample vector y.

FIG. 5 shows a frequency-domain response of the TEQ 121 with the TEQcoefficients and the TIR coefficients calculated by iteratively updatingthe optimum TEQ coefficient vector w_(opt) and the optimum TIRcoefficient vector b_(opt) obtained from Equations (9) and (10),respectively, where in particular, the ADSL transmission technology isadopted. As shown in FIG. 5, for downstream signals with the subcarriersin the upstream band (null frequency band) masked by zeros, theoptimized TEQ coefficients and the optimized TIR coefficientseffectively suppress the null frequency band. In comparison with FIG. 4,the null frequency band is effectively controlled, thereby improving theDMT transmission quality.

Therefore, by iteratively updating the TEQ coefficient vector and theTIR coefficient vector obtained from the modified cost function for theDMT transmission system with both time-domain equalization andfrequency-domain equalization, the DMT transmission quality can beimproved.

It will be apparent to those skilled in the art that variousmodifications and variations can be made to the structure of theinvention without departing from the scope or spirit of the invention.In view of the foregoing, it is intended that the invention covermodifications and variations of this invention provided they fall withinthe scope of the following claims.

What is claimed is:
 1. A method for discrete multitone (DMT)transmission, the method comprising: receiving a DMT signal from atransmission channel; passing the DMT signal through a time-domainequalizer (TEQ) to obtain an equalized DMT signal; passing the DMTsignal through a target impulse response (TIR) filter to obtain a TIRsignal; obtaining a mean square error (MSE) of an error signal from theequalized DMT signal and the TIR signal; and iteratively updating a TEQcoefficient vector of the TEQ based on the MSE of the error signal, afrequency kernel matrix corresponding to the TEQ and a frequency kernelmatrix corresponding to the TIR filter; wherein the frequency kernelmatrix Ω_(w) corresponding to the TEQ is determined asΩ_(w) =w _(l)Ω_(wl) +w _(h)Ω_(wh), where Ω_(wl) is a low-frequencykernel matrix corresponding to the TEQ, Ω_(wh) is a high-frequencykernel matrix corresponding to the TEQ, and w_(l) and w_(h) areweighting factors of the low-frequency kernel matrix Ω_(wl) and thehigh-frequency kernel matrix Ω_(wh), respectively.
 2. The method ofclaim 1, wherein the frequency kernel matrix Ω_(b) corresponding to theTIR filter is determined as:Ω_(b) =w′ _(l)Ω_(bl) +w′ _(h)Ω_(bh), where Ω_(bl) is a low-frequencykernel matrix corresponding to the TIR filter, Ω_(bh) is ahigh-frequency kernel matrix corresponding to the TIR filter, and w′_(l)and w′_(h) are weighting factors of the low-frequency kernel matrixΩ_(bl) and the low-frequency kernel matrix Ω_(bh), respectively.
 3. Themethod of claim 1, wherein the TEQ coefficient vector w is iterativelyupdated by the following equation:w=(R _(yy) +kΩ _(w))⁻¹ R _(yx) b, where R_(yy) is an autocorrelationmatrix of a sample vector y of the equalized DMT signal, k is aweighting factor of the frequency kernel matrix Ω_(w), R_(yx) is across-correlation matrix between the sample vector y and a sample vectorx of the DMT signal, and b is a TIR coefficient vector of the TIRfilter.
 4. The method of claim 1, further comprising: iterativelyupdating a TIR coefficient vector of the TIR filter based on the MSE ofthe error signal, the frequency kernel matrix corresponding to the TEQand the frequency kernel matrix corresponding to the TIR filter.
 5. Themethod of claim 4, wherein the TIR coefficient vector b is iterativelyupdated by the following equation:b=(R _(xx) +kΩ _(b))⁻¹ R _(xy) w, where R_(xx) is an autocorrelationmatrix of a sample vector x of the DMT signal, Ω_(b) is the frequencykernel matrix corresponding to the TIR filter, k is a weighting factorof the frequency kernel matrix Ω_(b), R_(xy) is a cross-correlationmatrix between the sample vector x and a sample vector y of theequalized DMT signal, and w is the TEQ coefficient vector.
 6. The methodof claim 1, wherein the frequency kernel matrix Ω_(w) is obtained byperforming an integration operation on a continuous frequency kernelmatrix variable with respect to a frequency range of a stopband of theTEQ.
 7. The method of claim 6, wherein the continuous frequency kernelmatrix variable is: $\begin{bmatrix}1 & {\cos(\omega)} & {\cos\left( {2\omega} \right)} & \ldots & {\cos\left\lbrack {\left( {N_{w} - 1} \right)\omega} \right\rbrack} \\{\cos(\omega)} & 1 & {\cos(\omega)} & \ldots & {\cos\left\lbrack {\left( {N_{w} - 2} \right)\omega} \right\rbrack} \\{\cos\left( {2\omega} \right)} & {\cos(\omega)} & 1 & \ldots & {\cos\left\lbrack {\left( {N_{w} - 3} \right)\omega} \right\rbrack} \\\vdots & \vdots & \vdots & \ddots & \vdots \\{\cos\left\lbrack {\left( {N_{w} - 1} \right)\omega} \right\rbrack} & {\cos\left\lbrack {\left( {N_{w} - 2} \right)\omega} \right\rbrack} & {\cos\left\lbrack {\left( {N_{w} - 3} \right)\omega} \right\rbrack} & \ldots & 1\end{bmatrix},$ where N_(w) is a TEQ length of the TEQ, and ω is thenormalized frequency of the stopband from π/2 to π.
 8. The method ofclaim 1, wherein the frequency kernel matrix corresponding to the TIRfilter is a summation operation, on a discrete frequency kernel matrixvariable with respect to a plurality of tones in a stopband of the TEQ.9. The method of claim 8, wherein the discrete frequency kernel matrixvariable is: $\quad{\begin{bmatrix}1 & {\cos\left( \frac{2\pi\; k}{N} \right)} & {\cos\left( \frac{4\pi\; k}{N} \right)} & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 1} \right)\pi\; k}{N} \right\rbrack} \\{\cos\left( \frac{2\pi\; k}{N} \right)} & 1 & {\cos\left( \frac{2\pi\; k}{N} \right)} & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 2} \right)\pi\; k}{N} \right\rbrack} \\{\cos\left( \frac{4\pi\; k}{N} \right)} & {\cos\left( \frac{2\pi\; k}{N} \right)} & 1 & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 3} \right)\pi\; k}{N} \right\rbrack} \\\vdots & \vdots & \vdots & \ddots & \vdots \\{\cos\left\lbrack \frac{2\left( {N_{b} - 1} \right)\pi\; k}{N} \right\rbrack} & {\cos\left\lbrack \frac{2\left( {N_{b} - 2} \right)\pi\; k}{N} \right\rbrack} & {\cos\left\lbrack \frac{2\left( {N_{b} - 3} \right)\pi\; k}{N} \right\rbrack} & \ldots & 1\end{bmatrix},}$ where N is a DMT signal length, N_(b)=CP+1, and CP is acyclic prefix length.
 10. An apparatus for discrete multitone (DMT)transmission, the apparatus comprising: a time-domain equalizer (TEQ)configured to pass a DMT signal therethrough to obtain an equalized DMTsignal; a target impulse response (TIR) filter configured to pass theDMT signal therethrough to obtain a TIR signal; an adder configured togenerate an error signal from the equalized DMT signal and the TIRsignal; and a processor configured to perform operations comprising:obtaining a mean square error (MSE) of an error signal from theequalized DMT signal and the TIR signal; determining a frequency kernelmatrix Ω_(w) corresponding to the TEQ from the following equation:Ω_(w) =w _(l)Ω_(wl) +w _(h)Ω_(wh), where Ω_(wl) is a low-frequencykernel matrix corresponding to the TEQ, Ω_(wh) is a high-frequencykernel matrix corresponding to the TEQ, and w_(l) and w_(h) areweighting factors of the low-frequency kernel matrix Ω_(wl) and thehigh-frequency kernel matrix Ω_(wh), respectively; and iterativelyupdating a TEQ coefficient vector of the TEQ based on the MSE of theerror signal, the frequency kernel matrix corresponding to the TEQ and afrequency kernel matrix corresponding to the TIR filter.
 11. Theapparatus of claim 10, wherein the processor is configured to performoperations comprising: determining the frequency kernel matrix Ω_(b)corresponding to the TIR filter from the following equation:Ω_(b) =w′ _(l)Ω_(bl) +w′ _(h)Ω_(bh), where Ω_(bl) is a low-frequencykernel matrix corresponding to the TIR filter, Ω_(bh) is ahigh-frequency kernel matrix corresponding to the TIR filter, and w′_(l)and w′_(h) are weighting factors of the low-frequency kernel matrixΩ_(bl) and the low-frequency kernel matrix Ω_(bh), respectively.
 12. Theapparatus of claim 10, wherein the processor is configured to performoperations comprising: iteratively updating the TEQ coefficient vector wby the following equation:w=(R _(yy) +kΩ _(w))⁻¹ R _(yx) b, where R_(yy) is an autocorrelationmatrix of a sample vector y of the equalized DMT signal, k is aweighting factor of the frequency kernel matrix Ω_(w), R_(yx) is across-correlation matrix between the sample vector y and a sample vectorx of the DMT signal, and b is a TIR coefficient vector of the TIRfilter.
 13. The apparatus of claim 10, wherein the processor isconfigured to perform operations comprising: iteratively updating a TIRcoefficient vector of the TIR filter based on the MSE of the errorsignal, the frequency kernel matrix corresponding to the TEQ and thefrequency kernel matrix corresponding to the TIR filter.
 14. Theapparatus of claim 13, herein the processor is configured to performoperations comprising: iteratively updating the TIR coefficient vector bby the following equation:b=(R _(xx) +kΩ _(b))⁻¹ R _(xy) w, where R_(xx) is an autocorrelationmatrix of a sample vector x of the DMT signal, Ω_(b) is the frequencykernel matrix corresponding to the TIR filter, k is a weighting factorof the frequency kernel matrix Ω_(b), R_(xy) is a cross-correlationmatrix between the sample vector x and a sample vector y of theequalized DMT signal, and w is the TEQ coefficient vector.
 15. Theapparatus of claim 10, herein the processor is configured to performoperations comprising: determining the frequency kernel matrix Ω_(w) byperforming an integration operation on a continuous frequency kernelmatrix variable with respect to a frequency range of a stopband of theTEQ.
 16. The apparatus of claim 15, wherein the continuous frequencykernel matrix variable is: $\begin{bmatrix}1 & {\cos(\omega)} & {\cos\left( {2\omega} \right)} & \ldots & {\cos\left\lbrack {\left( {N_{w} - 1} \right)\omega} \right\rbrack} \\{\cos(\omega)} & 1 & {\cos(\omega)} & \ldots & {\cos\left\lbrack {\left( {N_{w} - 2} \right)\omega} \right\rbrack} \\{\cos\left( {2\omega} \right)} & {\cos(\omega)} & 1 & \ldots & {\cos\left\lbrack {\left( {N_{w} - 3} \right)\omega} \right\rbrack} \\\vdots & \vdots & \vdots & \ddots & \vdots \\{\cos\left\lbrack {\left( {N_{w} - 1} \right)\omega} \right\rbrack} & {\cos\left\lbrack {\left( {N_{w} - 2} \right)\omega} \right\rbrack} & {\cos\left\lbrack {\left( {N_{w} - 3} \right)\omega} \right\rbrack} & \ldots & 1\end{bmatrix},$ where N_(w) is a TEQ length of the TEQ, and ω is thenormalized frequency of the stopband from π/2 to π.
 17. The apparatus ofclaim 10, wherein the processor is configured to perform operationscomprising: determining the frequency kernel matrix corresponding to theTIR filter by performing a summation operation on a discrete frequencykernel matrix variable with respect to a plurality of tones in astopband of the TEQ.
 18. The apparatus of claim 17, wherein the discretefrequency kernel matrix variable is: $\begin{bmatrix}1 & {\cos\left( \frac{2\pi\; k}{N} \right)} & {\cos\left( \frac{4\pi\; k}{N} \right)} & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 1} \right)\pi\; k}{N} \right\rbrack} \\{\cos\left( \frac{2\pi\; k}{N} \right)} & 1 & {\cos\left( \frac{2\pi\; k}{N} \right)} & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 2} \right)\pi\; k}{N} \right\rbrack} \\{\cos\left( \frac{4\pi\; k}{N} \right)} & {\cos\left( \frac{2\pi\; k}{N} \right)} & 1 & \ldots & {\cos\left\lbrack \frac{2\left( {N_{b} - 3} \right)\pi\; k}{N} \right\rbrack} \\\vdots & \vdots & \vdots & \ddots & \vdots \\{\cos\left\lbrack \frac{2\left( {N_{b} - 1} \right)\pi\; k}{N} \right\rbrack} & {\cos\left\lbrack \frac{2\left( {N_{b} - 2} \right)\pi\; k}{N} \right\rbrack} & {\cos\left\lbrack \frac{2\left( {N_{b} - 3} \right)\pi\; k}{N} \right\rbrack} & \ldots & 1\end{bmatrix},$ where N is a DMT signal length, N_(b)=CP+1, and CP is acyclic prefix length.